Theorem cnvss 4839

Description: Subset theorem for converse. (Contributed by NM, 22-Mar-1998.)

Assertion

Ref	Expression
cnvss	⊢ (𝐴 ⊆ 𝐵 → ◡𝐴 ⊆ ◡𝐵)

Proof of Theorem cnvss

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		ssel 3177	. . . 4 ⊢ (𝐴 ⊆ 𝐵 → (⟨𝑦, 𝑥⟩ ∈ 𝐴 → ⟨𝑦, 𝑥⟩ ∈ 𝐵))
2		df-br 4034	. . . 4 ⊢ (𝑦𝐴𝑥 ↔ ⟨𝑦, 𝑥⟩ ∈ 𝐴)
3		df-br 4034	. . . 4 ⊢ (𝑦𝐵𝑥 ↔ ⟨𝑦, 𝑥⟩ ∈ 𝐵)
4	1, 2, 3	3imtr4g 205	. . 3 ⊢ (𝐴 ⊆ 𝐵 → (𝑦𝐴𝑥 → 𝑦𝐵𝑥))
5	4	ssopab2dv 4313	. 2 ⊢ (𝐴 ⊆ 𝐵 → {⟨𝑥, 𝑦⟩ ∣ 𝑦𝐴𝑥} ⊆ {⟨𝑥, 𝑦⟩ ∣ 𝑦𝐵𝑥})
6		df-cnv 4671	. 2 ⊢ ◡𝐴 = {⟨𝑥, 𝑦⟩ ∣ 𝑦𝐴𝑥}
7		df-cnv 4671	. 2 ⊢ ◡𝐵 = {⟨𝑥, 𝑦⟩ ∣ 𝑦𝐵𝑥}
8	5, 6, 7	3sstr4g 3226	1 ⊢ (𝐴 ⊆ 𝐵 → ◡𝐴 ⊆ ◡𝐵)

Syntax hints:   → wi 4   ∈ wcel 2167   ⊆ wss 3157  ⟨cop 3625   class class class wbr 4033  {copab 4093  ^◡ccnv 4662

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-ext 2178

This theorem depends on definitions:  df-bi 117  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-in 3163  df-ss 3170  df-br 4034  df-opab 4095  df-cnv 4671

This theorem is referenced by:  cnveq  4840  rnss  4896  relcnvtr  5189  funss  5277  funcnvuni  5327  funres11  5330  funcnvres  5331  foimacnv  5522  tposss  6304  structcnvcnv  12694  pw1nct  15647